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          Floyd's cycle-finding algorithm
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        <p>引用维基百科的定义：<a target="_blank" rel="noopener" href="https://en.wikipedia.org/wiki/Cycle_detection#Floyd's_Tortoise_and_Hare">Floyd’s cycle-finding algorithm</a> is a pointer algorithm that uses only two pointers, which move through the sequence at different speeds. It is also called the “tortoise and the hare algorithm”.</p>
<p>此算法主要应用于链表中，主要是判断链表是否有环以及环的起始位置。</p>
<a id="more"></a>

<p>这两个内容刚好对应于 LeetCode 的两道题，此随笔就以这两道题来讲解一下这个巧妙的算法。下面给出这两道题在 LeetCode 中的链接：</p>
<ul>
<li><a target="_blank" rel="noopener" href="https://leetcode-cn.com/problems/linked-list-cycle/">141. 环形链表</a></li>
<li><a target="_blank" rel="noopener" href="https://leetcode-cn.com/problems/linked-list-cycle-ii/">142. 环形链表 II</a></li>
</ul>
<p>下面的实例代码都用 JavaScript 实现。</p>
<h2 id="判断链表是否有环"><a href="#判断链表是否有环" class="headerlink" title="判断链表是否有环"></a>判断链表是否有环</h2><p>此算法的主要思想就是双指针，<em>hare</em> 指针一次两步，<em>tortoise</em> 一次一步，如果链表有环，那么 <em>hare</em> 总会追上 <em>tortoise</em>。</p>
<p><em>hare</em> 一次三步可以吗，应该是可以的，如果有环，它们总会相遇的。但是可能 <em>hare</em> 就要辛苦些， 绕环跑多几圈。</p>
<figure class="highlight javascript"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param <span class="type">&#123;ListNode&#125;</span> <span class="variable">head</span></span></span></span><br><span class="line"><span class="comment"> * <span class="doctag">@return <span class="type">&#123;boolean&#125;</span></span></span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">var</span> hasCycle = <span class="function"><span class="keyword">function</span>(<span class="params">head</span>) </span>&#123;</span><br><span class="line">  <span class="keyword">if</span> (!head) <span class="keyword">return</span> <span class="literal">false</span>;</span><br><span class="line"></span><br><span class="line">  <span class="keyword">let</span> tortoise = head;</span><br><span class="line">  hare = head;</span><br><span class="line">  <span class="keyword">while</span> (hare &amp;&amp; hare.next) &#123;</span><br><span class="line">    hare = hare.next.next;</span><br><span class="line">    tortoise = tortoise.next;</span><br><span class="line">    <span class="keyword">if</span> (hare === tortoise) <span class="keyword">return</span> <span class="literal">true</span>;</span><br><span class="line">  &#125;</span><br><span class="line">  <span class="keyword">return</span> <span class="literal">false</span>;</span><br><span class="line">&#125;;</span><br></pre></td></tr></table></figure>

<h2 id="链表有环，判断链表环的起始位置"><a href="#链表有环，判断链表环的起始位置" class="headerlink" title="链表有环，判断链表环的起始位置"></a>链表有环，判断链表环的起始位置</h2><p>使用上面的算法，我们可以判断 <em>链表是否有环</em>。</p>
<p>如果有环，<em>hare</em> 和 <em>tortoise</em> 总会相遇，假设 <em>tortoise</em> 跑了 <code>k</code> 步，那么 <em>hare</em> 就跑了 <code>2k</code> 步，因为 hare 的速度是 tortoise 的两倍。</p>
<p>那么环的长度就是 <code>k</code>。</p>
<p>此时我们再假设 <em>相遇点</em> 到 <em>环的起点</em> 为 <code>m</code>，那么链表的起点到环的起点就是 <code>k - m</code> 的长度，相遇点再继续前进 <code>k - m</code> 步也会到环的起点，因为环的长度就是 <code>k</code> 嘛。</p>
<p>那么我们可以让 tortoise 和 hare，一个重回到链表的起点，一个从当前相遇点开始同步前进，一次一步。那么走了 <code>k - m</code> 步后它们就会相遇。</p>
<figure class="highlight javascript"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">// 这里让 tortoise 回到起点</span></span><br><span class="line"><span class="keyword">while</span> (hare != tortoise) &#123;</span><br><span class="line">  hare = hare.next;</span><br><span class="line">  tortoise = tortoise.next;</span><br><span class="line">&#125;</span><br><span class="line"><span class="keyword">return</span> hare;</span><br></pre></td></tr></table></figure>

<p>完整的代码如下：</p>
<figure class="highlight javascript"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment">/**</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param <span class="type">&#123;ListNode&#125;</span> <span class="variable">head</span></span></span></span><br><span class="line"><span class="comment"> * <span class="doctag">@return <span class="type">&#123;ListNode&#125;</span></span></span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"><span class="keyword">var</span> detectCycle = <span class="function"><span class="keyword">function</span>(<span class="params">head</span>) </span>&#123;</span><br><span class="line">  <span class="keyword">if</span> (!head) <span class="keyword">return</span> <span class="literal">null</span>;</span><br><span class="line">  <span class="keyword">let</span> hare = head;</span><br><span class="line">  <span class="keyword">let</span> tortoise = head;</span><br><span class="line"></span><br><span class="line">  <span class="comment">//找到相遇点</span></span><br><span class="line">  <span class="keyword">while</span> (hare &amp;&amp; hare.next) &#123;</span><br><span class="line">    hare = hare.next.next;</span><br><span class="line">    tortoise = tortoise.next;</span><br><span class="line">    <span class="keyword">if</span> (hare === tortoise) <span class="keyword">break</span>;</span><br><span class="line">  &#125;</span><br><span class="line"></span><br><span class="line">  <span class="comment">// 没有相遇点，也就是没有环，直接返回即可</span></span><br><span class="line">  <span class="comment">// 无环情况下，且结点数为奇数个时，hare指向最后一个结点</span></span><br><span class="line">  <span class="keyword">if</span> (hare != tortoise) <span class="keyword">return</span> <span class="literal">null</span>;</span><br><span class="line"></span><br><span class="line">  <span class="comment">//让tortoise指针回到开头，两指针重新同步前进，它们就会在环的起点相遇</span></span><br><span class="line">  tortoise = head;</span><br><span class="line">  <span class="keyword">while</span> (tortoise !== hare) &#123;</span><br><span class="line">    hare = hare.next;</span><br><span class="line">    tortoise = tortoise.next;</span><br><span class="line">  &#125;</span><br><span class="line">  <span class="keyword">return</span> hare;</span><br><span class="line">&#125;;</span><br></pre></td></tr></table></figure>

<h2 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h2><p><strong>Floyd’s cycle-finding algorithm</strong> 既是 <strong>tortoise and the hare algorithm</strong>，也是 <strong>双指针的快慢指针法</strong>。</p>
<p>此算法主要应用在链表的 <strong>判断链表是否有环</strong> 以及 <strong>如果链表有环，找出环的起始位置</strong>。</p>

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